Simultaneous decomposition of quaternion matrices involving η-Hermicity with applications

Abstract

Let R and Hm×n stand, respectively, for the real number field and the set of all m × n matrices over the real quaternion algebraH={a0+a1i+a2j+a3k|i2=j2=k2=ijk=−1,a0,a1,a2,a3∈R}.For η ∈ {i, j, k}, a real quaternion matrix A∈Hn×n is said to be η-Hermitian if Aη*=A where Aη*=−ηA*η, and A* stands for the conjugate transpose of A, arising in widely linear modeling. We present a simultaneous decomposition for a set of nine real quaternion matrices involving η-Hermicity with compatible sizes: Ai∈Hpi×ti,Bi∈Hpi×ti+1, and Ci∈Hpi×pi, where Ci are η-Hermitian matrices, (i=1,2,3). As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an η-Hermitian solution to the system of coupled real quaternion matrix equations AiXiAiη*+BiXi+1Biη*=Ci,(i=1,2,3), and provide an expression of the general η-Hermitian solutions to this system. Moreover, we derive the rank bounds of the general η-Hermitian solutions to the above-mentioned system using ranks of the given matrices Ai, Bi, and Ci as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper.